OFFSET
1,1
COMMENTS
The arithmetic mean (1/(n+1))*Sum_{k=0..n} a(k) converges to Pi. What is effectively the same: the Cesaro limit (C1) of a(n) is Pi. - Hieronymus Fischer, Jan 31 2006
A word that is uniformly recurrent, but not morphic. - N. J. A. Sloane, Jul 14 2018
REFERENCES
G. H. Hardy, Divergent Series, Oxford 1979.
Zeller, K. and Beekmann, W., Theorie der Limitierungsverfahren. Springer Verlag, Berlin, 1970.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..2000
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807, Nov 29 2017.
FORMULA
a(n) = A115790(n) + 3. - Michel Marcus, Jul 15 2013
EXAMPLE
a(6)=3 because 7*Pi = 21.99..., 6*Pi = 18.84..., so a(6) = 21 - 18;
a(7)=4 because 8*Pi = 25.13..., 7*Pi = 21.99..., so a(7) = 25 - 21.
MATHEMATICA
Differences[Floor[Pi Range[120]]] (* Harvey P. Dale, Jul 02 2021 *)
PROG
(PARI) j=[]; for(n=1, 150, j=concat(j, floor( (n+1) * Pi) - floor(n * Pi))); j
(PARI) { default(realprecision, 50); for (n=1, 2000, write("b063438.txt", n, " ", floor((n + 1)*Pi) - floor(n*Pi)) ) } \\ Harry J. Smith, Aug 21 2009
(PARI) a(n) = floor((n+1)*Pi) - floor(n*Pi) \\ Michel Marcus, Jul 15 2013
CROSSREFS
First differences of A022844.
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.
KEYWORD
easy,nonn
AUTHOR
Jason Earls, Jul 24 2001
EXTENSIONS
Offset in b-file and second PARI program corrected by N. J. A. Sloane, Aug 31 2009
Entry revised by N. J. A. Sloane, Jan 07 2014
STATUS
approved